Belmans–Smirnov's non-Hochschild-globality conjecture for isotropic Grassmannians

Let $X$ be an isotropic Grassmannian, meaning a Grassmannian parametrizing subspaces isotropic for the relevant symplectic or orthogonal form. A variety $X$ is Hochschild global if

$$H^i(X,\Lambda^j\mathcal{T}_X)=0$$

for all $i>0$ and all $j$.

Belmans–Smirnov's conjecture. If $X$ is neither (co)minuscule nor (co)adjoint, then $X$ is not Hochschild global; equivalently, for some $i>0$ and some $j$,

$$H^i(X,\Lambda^j\mathcal{T}_X)\neq 0.$$

Hochschild globality was known for (co)minuscule and (co)adjoint isotropic Grassmannians. The conjecture was established for the nonspecial isotropic Grassmannians considered in the paper, with the possible exception of $\mathsf{OGr}(n-1,2n+1)$ for $n\geq 4$; thus the original claim is not uniformly resolved in the stated generality.